The number *googol* is the number $10^{100}$, represented in the decimal system as follows: 10,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000. The number was named by Milton Sirotta, the nine-year-old nephew of American mathematician Edward Kasner, and popularized in Kasner's 1940 book *Mathematics and the imagination*. ### Googol plex A *googol plex* is the number $10^{\text{googol}}$, or $10^{10^{100}}$. In general, an $x$-plex is $10^x$. ### Googol bang A 'googol bang' is the factorial of a googol, that is, the number $(10^{100})!$. In general, an "$x$-bang" means $x!$. ### Googol stack A 'googol stack' is the number $10^{10^{10^{.^{.^.}}}}$, where the number of levels in the stack is a googol. This can also be represented $10\uparrow\uparrow\text{googol}$ in the Knuth notation for iterated exponentials. ### The googol plex bang stack hierarchy The plex, bang and stack vocabulary enables one to name some very large numbers with ease, such as a *googol bang plex stack*, which is the exponential tower $10^{10^{\cdot^{\cdot^{^{10}}}}}$ of height $10^{(10^{100})!}$, or a googol stack bang stack bang, and so on. The *googol plex bang stack hierarchy* is the collection of numbers that can be named in this scheme, by a term starting with googol and having finitely many adjectival operands: bang, stack, plex, in any finite pattern, repetitions allowed. The initial segment of this hierarchy, using term with at most three adjectival operands, appears below: - googol stack stack stack - googol bang stack stack - googol plex stack stack - googol stack bang stack - googol stack plex stack - googol stack stack bang - googol stack stack plex - googol stack stack - - all expressions involving at most one stack, and up to googol-2 many plexes and bangs appear here - - googol bang bang stack - googol bang plex stack - googol plex bang stack - googol plex plex stack - googol bang stack bang - googol bang stack plex - googol bang stack - googol plex stack bang - googol plex stack plex - googol plex stack - googol stack bang bang - googol stack bang plex - googol stack plex bang - googol stack plex plex - googol stack bang - googol stact plex - googol stack - - all terms involving up to googol-2 many bangs and plexes appear here - - googol bang bang bang - googol bang bang plex - googol bang plex bang - googol bang plex plex - googol plex bang bang - googol plex bang plex - googol plex plex bang - googol plex plex plex - googol bang bang - googol bang plex - googol plex bang - googol plex plex - googol bang - googol plex - googol A sorting algorithm for all such names in the hierarchy involving fewer than googol-2 many terms is provided by <a href="http://math.stackexchange.com/a/92661/413" class="external text">this math.stackexchange answer</a>.